Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and sinh(t) respectively. Hyperbolic functions are used to express the angle of parallelism in hyperbolic geometry .

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use the natural exponential function e x. And are not the same as sin(x) and cos(x), but a little bit similar: sinh vs sin. cosh vs cos. Catenary

$\text{cosh}\ x\ \text{cosh}\ y = \frac12(\text{cosh}(x + y) + \text{cosh} (x - y))$ $\text{sinh}\ x\ \text{cosh}\ y = \frac12(\text{sinh}(x + y) + \text{sinh} (x - y))$ Expression of hyperbolic functions in terms of others

Illustrated definition of Cosh: The Hyperbolic Cosine Function. cosh(x) (esupxsup esupminusxsup) 2 Dont confuse it with...

There are six hyperbolic functions are sinh x, cosh x, tanh x, coth x, sech x, csch x. In this article, we will define these hyperbolic functions and their properties, graphs, identities, derivatives, etc. along with some solved examples.

The hyperbolic cosine function, or cosh(x), is one of the various hyperbolic functions. Its evaluation involves Euler’s number e . For an input x , the hyperbolic cosine’s output is the sum of e to the power x and e to the power minus x, divided by 2.

Cosh, short for hyperbolic cosine, is a mathematical function that is part of the family of hyperbolic functions. It is closely related to the hyperbolic sine function and is used extensively in the study of calculus, particularly in the context of the calculus of hyperbolic functions.

The two basic hyperbolic functions are sinh and cosh. sinh(x) = (e x − e −x)/2 cosh(x) = (e x + e −x)/2 (From those two we also get the tanh, coth, sech and csch functions.) Here you see how sinh compares to the sine function and cosh compares to the cosine function:

Cosh is the hyperbolic cosine function, which is the hyperbolic analogue of the Cos circular function used throughout trigonometry. It is defined for real numbers by letting be twice the area between the axis and a ray through the origin intersecting the unit hyperbola .

So for some suitable \(t\), \(\cosh t\) and \(\sinh t\) are the coordinates of a typical point on the hyperbola. In fact, it turns out that \(t\) is twice the area shown in the first graph of Figure \(\PageIndex{2}\).